Stochastic Processes at CEU -- winter semester 2018/19

Homeworks to hand in by the 19th of February: 1.3; 1.4; 1.7; 1.9; 1.10

No. of Credits: 3
No. of ECTS credits: 6
Time Period of the course: Winter Semester
Prerequisites: Probability 1
Course Level: PhD
Syllabus: will be here.
Classes: Tuesdays from 09:00 in room 301.
Course Coordinator: Imre Péter Tóth.

Planned schedule
week #whentopicremark
week 12019.01.08Introduction, overview. Finite dimensional distributions and construction of stochastic processes. Kolmogorov extension theorem.
week 22019.01.15Classification of stochastic processes, examples. Poisson process, Wiener process. Markov property, finite Markov chains.
week 32019.01.22Mixing and ergodicity of Markov chains. Countable Markov chains. Transience, recurrence, positive recurrence. Random walks on Z^d, Pólya's theorem.
week 42019.01.29Random walks on Z. The reflection principle and applications. Distribution of the maximum, arcsine law, return times, local time.CLASS DELAYED to the next three weeks
week 52019.02.05Discrete time martingales. Branching processes. Barabási-Albert graph model, preferential attachment.CLASS LONGER with 35 minutes + a break
week 62019.02.12Discrete time stochastic integration. Discrete Black-Scholes formula.CLASS LONGER with 35 minutes + a break
week 72019.02.19Brownian motion, Wiener process. Construction(s), regularity properties, scaling properties.CLASS LONGER with 35 minutes + a break
week 82019.02.26Markov chains in continuous time. Infinitesimal generator, exponential semigroup. Examples.
week 92019.03.05Filtrations and stopping times. Markov processes and martingales in continuous time.
week 102019.03.12Ito's stochastic integral. Motivation, definition, basic properties.
week 112019.03.19Itô processes and the Itô formula. Examples of stochastic differential equations.
week 122019.03.26Itô representation theorem, Martingale representation theorem.

Suggested literature:
Grading rules:
There will be three homework assignments -- roughly once in every four weeks -- worth 20% of the total score. A midterm exam focused on problem solving will be worth 30%, while the final exam, focused on theory will be worth 50%.

Requirements for "audit":
regular participation in class and a short oral account of the concepts and phenomena learned.